Two-variable Logic with a Between Predicate

TL;DR

This paper introduces an extension of two-variable first-order logic with a between predicate, analyzing its expressive power, complexity, and algebraic properties, revealing it is less expressive than full FO[<].

Contribution

It defines and studies a new two-variable logic with a between predicate, providing complexity bounds and algebraic characterizations of its expressiveness.

Findings

Matching bounds for satisfiability complexity.

An algebraic necessary and sufficient condition for expressibility.

The logic is less expressive than full FO[<].

Abstract

We study an extension of FO^2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, `the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables. We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We also give an effective necessary condition, in terms of the syntactic monoid of a regular language, for a property to be expressible in this logic. We show that this condition is also sufficient for words over a two-letter alphabet. This algebraic analysis allows us us to prove, among other things, that our new logic has strictly less expressive power than full first-order logic FO[<].